Equality of opportunity versus equality of opportunity sets∗

Erwin Ooghe†, Erik Schokkaert‡, Dirk Van de gaer§

December 18, 2003

Abstract

We characterize two different approaches to the idea of equality of opportunity.Roemer’s social ordering is motivated by a concern to compensate for the effectsof certain (non-responsibility) factors on outcomes. Van de gaer’s social orderingis concerned with the equalization of the opportunity sets to which people haveaccess. We show how different invariance axioms open the possibility to go beyondthe simple additive specification implied by both rules. This offers scope for abroader interpretation of responsibility-sensitive egalitarianism.

JEL-classification: D64

Keywords: equality of opportunity, opportunity sets, responsibility-sensitive egal-itarianism

1 Introduction

The traditional idea of equality of opportunity has been interpreted in two differentways in the recent literature. A first approach starts from the premise that individualoutcomes are determined by two types of factors: compensation factors (such as innateabilities, handicaps, called “type” in the sequel) and responsibility factors (such asambition, preferences, called “effort” in the sequel).1 The basic idea of equality ofopportunity then is to equalize individual outcomes to the extent that they reflectdifferences in “type”, while allowing for different outcomes whenever they are due to“effort”. This focus on compensation of outcomes is prominent in the larger part of theaxiomatic literature –see, e.g., Bossert (1995), Fleurbaey (1994, 1995a, 1995b), Bossertand Fleurbaey (1996), Iturbe-Ormaetxe (1997), Maniquet (1998, 2002). It is also thebasic inspiration of Roemer’s rule (Roemer, 1993, 1998, 2002).

∗We thank two anonymous referees for their comments on an earlier version. We also wish to acknowl-edge financial support from the Belgian Programme on Interuniversity Poles of Attraction, initiated bythe Belgian State, Prime Minister’s Office.

†Postdoctoral Fellow of the Fund for Scientific Research - Flanders. Center for Economic Studies,KULeuven, Naamsestraat, 69, B-3000 Leuven, Belgium. e-mail: erwin.ooghe@econ.kuleuven.ac.be.

‡Center for Economic Studies, KULeuven, Naamsestraat, 69, B-3000 Leuven, Belgium. e-mail:erik.schokkaert@econ.kuleuven.ac.be.

§Vakgroep Sociale Economie, UGent, Hoveniersberg 24, B-9000 Gent, Belgium. e-mail:dirk.vandegaer@rug.ac.be.

1Roemer (2002) calls these “arbitrary” and “responsible” factors.

1

A second approach to equality of opportunity focuses on the opportunity set to whichpeople have access, and tries to make these sets as equal as possible –see, e.g., Kranich(1996), and Ok and Kranich (1998). Compensation is defined in terms of opportunitysets. The concern for responsibility implies that only the set matters, while individualsremain responsible for their choice. A rule which starts from this inspiration has beenproposed by Van de gaer (1993) and is further explored in Bossert, Fleurbaey and Vande gaer (1999).Of course the two approaches are related, since the opportunity set to which a personwith a given type has access gives rise to the vector of possible outcomes that he canobtain for different values of effort. Yet their basic inspiration is different.Roemer’s approach has been very popular in the recent past and it has resulted in aseries of thought provoking empirical applications (Llavador and Roemer, 2001, Roemeret al., 2002, Roemer, 2002). As a matter of fact, in his recent “progress report”, Roemer(2002) explicitly criticizes the alternative “opportunity set” approach, claiming that ithas not resulted in any empirical work because it remains too abstract and fails to takesufficient cues from popular and long-standing conceptions of equality of opportunity.This criticism is certainly not valid for the Van de gaer rule. In many cases, both theRoemer rule and the Van de gaer rule lead to exactly the same policy prescriptions.This happens to be true in all the empirical applications referred to earlier. In caseswhere the rules differ, the Van de gaer rule is even computationally simpler than theRoemer rule, at least in a continuous setting, because the latter has to integrate thelower counter set of the different types. Empirical applicability therefore cannot be anargument to discard the former.In this paper, we want to explore in more detail the differences between the Roemer ruleand the Van de gaer rule. We briefly review both rules in section 2. We illustrate howRoemer’s rule is an example of the CO (compensating outcomes) approach, while Vande gaer’s rule is an example of the CS (compensating sets) approach. Moreover, bothrules assume an additive aggregation procedure. In section 3, we introduce notation andpresent both basic rules, together with some extensions. Section 4 discusses the axiomsand section 5 characterizes and discusses the basic versions of the Roemer rule and theVan de gaer rule. The axioms used in the characterization reveal the strong and weakpoints of both. They also yield a better insight into the implications of the additiveaggregation rule underlying both social orderings. In section 6 we show how relaxingthis additivity assumption paves the way for a characterization of some alternative rules(also proposed in Kolm, 2001). Section 7 concludes.

2 Two alternative rules

To set the scene, let us first briefly summarize both approaches in a finite setting. Let Xbe the set of social options, e.g. the different policy prescriptions. Individuals who arehomogeneous with respect to compensation factors are of the same “type”; we gatherthe different types i = 1, . . . ,m in a set M . In Roemer’s writings, the degree of effortis identified on the basis of the rank order of an individual within the distribution ofoutcomes of his type. We will avoid this debatable identification assumption.2 Individu-

2Roemer proposes to identify effort on the basis of the quantiles of the outcome distribution for eachtype. For a critical assessment of this quantile hypothesis, see Fleurbaey (1998) and Kolm (2001). For

2

als who are homogeneous with respect to responsibility characteristics have exerted thesame “effort”; the different effort levels j = 1, . . . , n are collected in a set N . To simplifyour exposition, we assume that each type-effort couple (i, j) ∈ M × N is representedby exactly one individual; we abbreviate ij ∈MN . Later on, we mention how to adaptthe rules when relaxing this assumption. Outcomes are defined by the social option,the type, and the effort of the individual. Let U = U11 , U

21 , . . . , U

nm be a profile of

real-valued outcome functions U ji , one for each type-effort couple ij, defined as

U ji : X → R (or R++) : x→ U ji (x) .

To illustrate both approaches, we consider an economy with two types and five effortlevels. Figure 1 presents the outcomes of both types under a certain social option x forthe different effort levels, i.e. the vectors Ui (x) = U1i (x) , . . . , U5i (x) , i = 1, 2.

0

2

4

6

8

10

12

14

16

1 2 3 4 5

effort

outc

om

e

type 1 type 2

Figure 1: Relation between effort and outcome for two types

Either one tries to equalize the outcomes of different types at the same effort level: thisis the CO (compensating outcomes) approach to which Roemer’s rule belongs. Definethe social option xj which maximizes the minimal outcome of the different types ateffort level j, i.e.

xj = arg maxx∈X

mini∈M

U ji (x) .

It is rather unlikely that xj will be the same for all effort levels. Therefore, it is nec-essary to look for an adequate aggregation rule over the different effort levels. Roemerproposes –without much justification– a simple additive specification for this purpose.Roemer’s social objective becomes

maxx∈X

j∈Nmini∈M

U ji (x) . (1)

Figure 2 presents the minimal outcomes for each effort level in black; the Roemer rulewants to maximize the sum of these outcomes.

a strong defence, see Roemer (2002).

3

0

2

4

6

8

10

12

14

16

1 2 3 4 5

effort

outc

om

e

type 1 type 2

Figure 2: Roemer’s sum-of-mins rule

Or one tries to equalize the opportunity sets of the different types; this is in line withthe CS (compensating sets) approach to which Van de gaer’s rule belongs. The Vande gaer rule follows Bossert (1997), who values an opportunity set solely on the basisof the corresponding outcomes. Here again one has to define how to aggregate theseoutcomes. Again without too much justification, Van de gaer proposes a simple additivespecification: the value of the opportunity set of type i under option x becomes

j∈NU ji (x) .

Equalizing opportunity sets boils down to equalizing their valuation; the maximin-rulereadily suggests itself. Van de gaer’s rule maximizes the value of the opportunity set ofthe most disadvantaged type:

maxx∈X

mini∈M

j∈NU ji (x) . (2)

Figure 3 shows the outcomes of the most disadvantaged (lowest sum of outcomes) –heretype 1– in black; the Van de gaer rule wants to maximize the sum of these outcomes.Comparing (1) and (2) shows that the only formal difference is the interchange of themin and the sum operator.3 If the same type is the least advantaged at each effort level,both criteria lead to the same policy. Although formally similar, the basic intuitionbehind both rules –CO versus CS– is very different.4 We provide an example wherethe ranking of two social options is different between both rules.

3 In fact, Roemer (2002) presents the Van de gaer-rule as an alternative compromise procedure incase the options xj , defined in equation (??), are different for some effort levels. Roemer has no strongpreference for any of these alternatives over the other. In this spirit, our paper can be read as acomparison of different compromise procedures in the Roemer-approach. However, we feel that thedifferences are deeper than suggested by this interpretation.

4Kolm (2001) suggests to interpret Roemer’s criterion via the normative concept “desert” and Vande gaer’s in terms of its dual, “merit”.

4

0

2

4

6

8

10

12

14

16

1 2 3 4 5

effort

outc

om

e

type 1 type 2

Figure 3: Van de gaer’s min-of-sums rule

0

2

4

6

8

10

12

14

16

1 2 3 4 5

effort

outc

om

e

type 1 type 2

Figure 4: Roemer versus Van de gaer

Figure 4 illustrates the incompatibility of both approaches. It presents the opportunitysets of two types (bullets and diamonds) over five effort levels in black; except for thelowest effort level, type 1 is everywhere better off. Now, suppose we can redistributeoutcomes for the lowest effort level only, such that outcome levels after the transfer (inwhite) coincide. According to the Roemer rule, this policy is an improvement, althoughnow type 1 dominates type 2 after the change; thus, the CO-approach neglects opportu-nity set considerations. The Van de gaer rule, would prefer the inverse policy; thus, theCS-approach may increase the outcome differences for individuals who exert the sameeffort.This example suggests that there is really a basic difference between the Roemer ruleand the Van de gaer rule –a difference which goes deeper than a simple normalization.

5

At the same time, however, both rules have analogous weaknesses. Consider a situationwith only one type. Both the Roemer rule and the Van de gaer rule will then evaluatedifferent policies in a utilitarian fashion, i.e. as the sum of the outcomes at the differenteffort levels. It has been argued (see, e.g., Fleurbaey, 1998) that this goes against theintuition of the idea of equal opportunity because the government should not interveneand should simply accept the laissez-faire outcomes, if the differences between outcomesresult only from differences in effort.It is true that this is the usual interpretation of equality of opportunity. At the sametime, however, a broader idea of responsibility-sensitivity does not necessarily implyan absolute respect for laissez-faire. It is possible to take the position that the socialevaluation should focus on opportunities (and opportunity sets) rather than on outcomeswhile at the same time postulating a definite idea about what should be the optimalform of these sets. In the latter interpretation it seems useful to think about ways toformulate a more flexible approach than the sum-specification of both the Roemer ruleand the Van de gaer rule.The main aim of this paper is to further explore the basic differences between thestandard Roemer rule and the Van de gaer rule (section 5), and to look for more flexibleextensions (section 6). Therefore, the next two sections (3 and 4) introduce the necessarynotation.

3 Social orderings

Notation We assume strong neutrality as defined by Roberts (1995) for his “ex-tensive” social choice framework. As a consequence, we can focus on a social ordering5

R over outcome vectors u = u11, u21, . . . , u

nm , rather than on an extensive social welfare

functional defined over profiles U = U11 , U21 , . . . , U

nm . For convenience, we will consider

outcome matrices:

u = uji =

u11 . . . un1. . . . . .

u1i . . . uji . . . uni. . . . . .u1m . . . unm

Row ui = u1i , . . . , u

ni represents the outcome vector for type i (the opportunities),

while column uj = uj1, . . . , ujm represents the outcome vector for effort level j. We

introduce three rank-operators. The first one ranks outcomes within each column of amatrix u in an increasing way; the resulting matrix equals u, with uj1 representing thelowest outcome for effort level j in the original matrix u and so on. The second onereranks the rows in an increasing way, on the basis of the sum of their outcomes; the

resulting matrix equals+u, with

+u1=

+u1

1, . . . ,+un

1 the opportunities of the type with

the lowest sum of outcomes in the original matrix u and so on. The final one alsoreranks the rows of a strictly positive matrix in an increasing way, but on the basis ofthe product of its outcomes;

xu is the matrix obtained from u in this way.

5An ordering is a complete and transitive binary relation. P and I denote the corresponding asym-metric and symmetric relation.

6

The Roemer rule and the Van de gaer rule We present Roemer and Van degaer’s proposal, slightly modified to satisfy the strong Pareto principle; we thereforereplace maximin by leximin. The Roemer rule can be rephrased as lexicographicallymaximizing the sum of the minimal outcomes at each effort level; uRv holds, if andonly if6

either, j∈N uji = j∈N v

ji , i = 1, . . . ,m ,

or, ∃s, 1 ≤ s ≤ m : j∈N uji = j∈N v

ji , i < s and j∈N u

js > j∈N v

js.

In the same vein, the Van de gaer rule can be reformulated as lexicographically max-imizing the minimal sum of outcomes for the different types; uRv holds, if and onlyif7 either, j∈N

+uj

i= j∈N+vj

i , i = 1, . . . ,m ,

or, ∃s, 1 ≤ s ≤ m : j∈N+uj

i= j∈N+vj

i , i < s and j∈N+uj

s> j∈N+vj

s .

The main focus of this paper will be on the characterization of these two basic socialorderings. However, as mentioned already in the previous section, the sum-componentin both rules has hardly been justified in the literature until now. As a by-productof our characterization we will better understand the consequences of this additivespecification. In section 6, we will characterize some alternative rules.

The multiplicative Roemer rule and the Van de gaer rule Instead of adding,we may also multiply outcomes over different effort levels. Restricting the domain tostrictly positive outcome matrices, the product-Roemer rule ranks uRv, if and only if:

either, j∈N uji = j∈N v

ji , i = 1, . . . ,m ,

or, ∃s, 1 ≤ s ≤ m : j∈N uji = j∈N v

ji , i < s and j∈N u

js > j∈N v

js.

The product-Van de gaer rule ranks uRv, if and only if either, j∈Nxuj

i= j∈Nxvj

i , i = 1, . . . ,m ,

or, ∃s, 1 ≤ s ≤ m : j∈Nxuj

i= j∈Nxvj

i , i < s and j∈Nxuj

s> j∈Nxvj

s .

Flexible Roemer rule and Van de gaer rule Finally, we introduce a Kolm-Atkinson-Sen (KAS) and Kolm-Pollak (KP) specification, which allows for a differentinequality aversion parameter to aggregate over types (via ρM ∈ R) and over effort (viaρN ∈ R). These specifications have been proposed by Kolm (2001). The KAS-Roemer-family and the KP-Roemer-family contain social orderings preferring u to v, if and only

6 If one allows for a different number nij of individuals with characteristics ij ∈MN , the mins haveto be weighted by proportions pj = i nij

ij nij.

7 If one allows for a different number nij of individuals with characteristics ij ∈MN , Van de gaer’srule has to be based on sums of outcomes, weighted by the conditional proportion pij/i =

nij

j nij. The

resulting comparison of two matrices has to be weighted in the same way.

7

if respectively8j∈N

ujρN

1ρN

≥j∈N

vjρN

1ρN

, with xj =i∈M

xjiρM

1ρM

, x = u, v ,

1

ρNlnj∈N

eρNuj ≥ 1

ρNlnj∈N

eρNvj, with xj =

1

ρMlni∈M

eρMxji , x = u, v.

The KAS-Van de gaer-family and KP-Van de gaer-family invert the aggregation steps.They contain social orderings preferring u to v, if and only if respectively:

i∈M(ui)

ρM

1ρM

≥i∈M

(vi)ρM

1ρM

, with xi =

j∈N

xjiρN

1ρN

, x = u, v ,

1

ρMlni∈M

eρMui ≥ 1

ρMlni∈M

eρMvi , with xi =1

ρNlnj∈N

eρNxji , x = u, v.

4 Properties

We focus on a social ordering defined over matrices in a domain D, with D equal toRm×n or Rm×n++ :

• UD (unrestricted domain).R is defined on D = Rm×n.

• PD (positive domain). R is defined on D = Rm×n++ .

Continuity is defined as:

• CON (continuity).For each u ∈ D; the better-than and worse-than sets {v ∈ D |vRu} and {v ∈ D |uRv}are closed.

Efficiency boils down to the strong Pareto principle, defined as:

• SP (strong Pareto).For each u ∈ D; if uji ≥ vji for all ij ∈MN , then uRv. If, in addition, there existsa kl ∈MN such that ulk > v

lk, then uPv.

The main differences between the CO-approach and the CS-approach are linked to sep-arability, impartiality and compensation. The columns in Table 1 present the differentaxioms within these three classes, while the rows indicate the two main approaches. Thesymbol + (resp. − ) indicates whether an axiom is typical (resp. atypical) for one ofboth approaches; the symbol • refers to axioms that will be used in the next section tocharacterize the Roemer rule and the Van de gaer rule.

8The KAS-specification is only well-defined for strictly positive outcome matrices. As usual, whenρ = 0 the KAS- and KP-specifications are equal to respectively a log-utilitarian and a utilitarian rule.

8

Table 1: Compensating outcomes versus compensating sets

separability (SE) between impartiality (SI) for compensation (C)effort (N) type (M) effort (N) type (M) (for type only)

SE∗N SEN SE∗M SEM SEM SI∗N SIN SI∗M SIM C DC

CO + + − + • − • • + + •CS − + + + + • + − • − •

Separability For reasons that will become clear later on, separability assumptionsplay only a minor role in the characterization of the basic Roemer rule and the Vande gaer rule (see table 1). They are more important to characterize the flexible exten-sions in section 6. In a CS-approach it seems straightforward to impose a separabilityrequirement between the different rows (opportunities of a type). Strong separabilitybetween types eliminates indifferent types, i.e. types with identical opportunities undertwo social options do not play a role:

• SE∗M (strong separability between types).For all u, v, u , v ∈ D and for all subsets G ⊆ M ; if uji = vji and u

ji = v ji for all

ij ∈ GN , while uji = u ji and vji = v ji for all ij ∈MN\GN , then uRv if and only

if u Rv .

In the same way we can define strong separability between effort (columns). This axiomeliminates indifferent effort levels, i.e. effort levels with identical outcome vectors. Thisaxiom is very intuitive for the CO approach.

• SE∗N (strong separability between effort).For all u, v, u , v ∈ D and for all subsets G ⊆ N ; if uji = vji and u

ji = v ji for all

ij ∈MG, while uji = u ji and vji = v ji for all ij ∈MN\MG, then uRv if and only

if u Rv .

For both separability requirements, we can also introduce a weaker version, which re-quires separability in case of specific subdomains. DM contains matrices where outcomesin each row are constant: effort has no effect on the outcomes of each type. DN consistsof matrices where outcomes in each column are constant: type has no effect on theoutcomes for each effort level.

• SEM (separability between types).SE∗M holds for all u, v, u , v ∈ DM .

• SEN (separability between effort).SE∗N holds for all u, v, u , v ∈ DN .

Similar to Ebert (1988), we could require separability between types, but only forcolumn-ordered matrices. Recall the rank operator ∼; we define

D = {v ∈ D |∃u ∈ D such that v = u} ,and we get:

9

• SEM (ordered separability between types).SE∗M holds for all u, v, u , v ∈ D.

For the characterization of the Roemer rule we will only need this last axiom SEM ; alsothe Van de gaer rule satisfies it.

Impartiality Much more important are the impartiality requirements. Suppes’indifference implies that the “names” of the types (resp. effort levels) do not matter.We will define it in two different strengths: we can permute rows (resp. columns), butstronger, we can also permute outcomes within each column (resp. row) separately. Letσ denote a permutation of a set. When reasoning within a CS-approach, the followingcombination of axioms seems reasonable:

• SIM (Suppes’ indifference for types).

For each u ∈ D and for all permutations σ on M ; uIu , with u = ujσ(i) .

• SI∗N (strong Suppes’ indifference for effort).For each u ∈ D and for each m-tuple (σ1, . . . ,σm) of permutations on N ; uIu ,

with u = uσi(j)i .

The first axiom SIM is a natural requirement from a CS perspective. It imposes thatthe different types should be treated in a symmetric way: their opportunity vectors canbe permuted without influencing the social evaluation. The second one, SI∗N , is also aplausible requirement if we look at the rows of the matrices as embodying the valuesof the opportunity sets to which different individuals have access. It imposes that onlythe whole vector of opportunities matters, not the specific link between each effort leveland the resulting outcome.On the other hand, in the CO-approach we can reasonably assume:

• SIN (Suppes’ indifference for effort).For each u ∈ D and for all permutations σ on N ; uIu , with u = u

σ(j)i .

• SI∗M (strong Suppes’ indifference for types).For each u ∈ D and for each n-tuple σ1, . . . ,σn of permutations on M ; uIu ,

with u = ujσj(i)

.

In the CO-setting, SI∗M is a very natural requirement. When comparing the outcomesfor different types at the same effort level, the identity of the types should not matter.SIN permutes entire columns, or the names of the effort levels do not matter.

Compensation In the CO approach, compensation requires equalization of theoutcomes of the different types who exerted the same effort. In the spirit of Hammond(1976), we define:

• C (compensation).For all u, v ∈ D and for all ij, kj ∈ MN ; if vjk > ujk ≥ uji > vji , while uhg = vhg forall other gh = ij, kj, then uRv.

10

In the CS-approach however, compensation focuses on the equalization of opportunityvectors. Valuing opportunity vectors is a difficult problem and we do not want to imposestrong assumptions about it at this stage. We therefore start from the simple idea thatan opportunity set is “better” than another if it gives rise to higher outcomes for alleffort levels. Given two types i and k, we write ui > uk to denote u

ji ≥ ujk, for all j in N ,

while for at least one l in N , we have uli > ulk; in this case, we say that type i (strictly)

dominates type j. Adding such a dominance requirement to the previous compensationaxiom gives us dominance compensation:9

• DC (dominance compensation).For all u, v ∈ D and for all ij, kj ∈ MN ; if vi > vk and vjk > ujk ≥ uji > vji , whileuhg = vhg for all other gh = ij, kj, then uRv.

The Van de gaer rule implies much more than only a dominance condition with respect tothe evaluation of opportunity sets. It incorporates the following utilitarian compensationprinciple:10

• UC (utilitarian compensation).For all u, v ∈ D and for all types i, k ∈ M ; if j∈N v

jk > j∈N u

jk ≥ j∈N u

ji >

j∈N vji , while u

hg = vhg for all gh /∈ iN ∪ kN , then uRv.

We do not use this stronger axiom for the characterization of the Van de gaer rule,but we show that it is implied by the combination of other axioms. For later reference,we finish by introducing the idea of utilitarian neutrality. If the sum of opportunitiesremains the same for all types, then we should rank the matrices as indifferent:

• UN (utilitarian neutrality).For all u, v ∈ D; if j∈N u

ji = j∈N v

ji for all i in M , then uIv.

5 Results

The Roemer rule and the Van de gaer rule Roemer’s CO-approach focuses onthe differences between the outcomes for different types at the same effort level, i.e. onthe inequality within the columns of the outcome matrix. It then seems natural torequire that the social evaluation should be invariant if all outcomes within the columnsare changed by the same absolute number or are changed in the same proportion sincesuch changes do not affect absolute or relative inequality, respectively. We will seethat the Roemer rule focuses on absolute inequality within a column. Roemer remainsagnostic, however, about the differences between the different columns. Therefore, thesocial evaluation does not depend on the differences between outcomes at different effortlevels.

9This differs from the approach taken by Ok and Kranich (1998) who advocate to define equaliza-tion of opportunity sets for a given “cardinality-based” ordering of all possible opportunity sets. Our“ordinality-based” notion of dominance compensation is the equivalent of “opportunity dominance fora policy” as introduced by Hild and Voorhoeve (2001).10The cartesian product {i} ×N is abbreviated as iN .

11

To formalize this idea, consider Σ, the domain of matrices α ∈ Rm×n with all elementsequal in a column; for all couples ij, kj inMN , we have αji = αjk. We define the followingaxiom:11

• Itntf (invariance axiom).For all u, v ∈ D and for all α ∈ Σ; uRv if and only if (u+ α)R (v + α) .

Notice that Itntf implies separability between effort levels SEN . The axiom Itntf means that

the social judgment does not change provided that the absolute differences in outcomesat each effort level, i.e. within each column, are preserved. At the same time, sincethe elements of matrix α may differ over different effort levels, differences in outcomesbetween the columns are irrelevant for the social evaluation. Within the CO-approach,the Itntf axiom forces us to focus on the inequality of outcomes, irrespective of the levelat which these inequalities are evaluated.The Itntf axiom is equally relevant in the CS-approach. Indeed, if the difference in out-comes for each effort level remains the same, then the difference in the valuation ofthe opportunity sets also remains the same. If one wants to focus on the inequalityof the sets, the α-transformation should therefore not lead to a change in the relativeevaluation of different opportunity sets.We are now ready to formulate our two basic characterization results:12

Proposition 1. For each row in the table, there is only one social ordering R whichsatisfies the axioms indicated by •; it is the rule in the first column. In addition, +(resp. − ) means that the rule satisfies (violates) the corresponding axiom:

UD Itntf SP DC SEM SI∗M SIM SI∗N SINRoemer • • • • • • + − •Van de gaer • • • • + − • • +

Discussion In the proof of proposition 1 (see appendix), we also show that allaxioms are independent: for each axiom we drop, we present a different rule whichsatisfies all axioms, except the dropped one. Note that the only separability assumptionused in the characterization of the Roemer rule is SEM . This axiom is also satisfied bythe Van de gaer rule, but it is redundant for its characterization. The reason for thelimited importance of separability axioms is that their potential role is taken over bythe invariance axiom Itntf and the equity axiom DC.13

More surprising is the fact that we use in both cases the same dominance compensationaxiom DC. The only substantial differences are to be found in the impartiality axioms.As we show next, they have strong implications.

11From a formal point of view, the invariance axiom Itntf is identical to a two-dimensional invarianceassumption, introduced in Ooghe and Lauwers (2003). It amounts to translation-scale measurability(T) combined with full comparability (F) between the different types (for a given effort level) andtranslation-scale measurability (T) and no comparability (N) between the different effort levels (for agiven type).12All the proofs are put together in the appendix.13Ooghe and Lauwers (2003) provide an alternative characterization of the Roemer-rule, via additional

separability requirements, combined with a weaker (minimal) equity requirement.

12

The proof of proposition 1 for the Van de gaer rule makes use of the following lemma1, showing that the combination of the invariance axiom Itntf with strong Suppes’ indif-ference between effort levels SI∗N implies utilitarian neutrality UN:

Lemma 1. If a social ordering R defined on D satisfies UD, Itntf and SI∗N , then it also

satisfies UN.

It is clear that UN makes sense only in the CS approach, where sets are measured by thesum of the outcomes of those that have the same compensation characteristic. Lemma1 shows that, in combination with UD and Itntf , the axiom SI∗N brings us already a longway in the direction of the Van de gaer rule. This becomes even more transparent whenwe impose also the dominance compensation axiom DC, because one can show

Lemma 2. If a social ordering R defined on D satisfies UD, Itntf , SI∗N and DC, then it

also satisfies UC.

The other impartiality requirement SI∗M plays an equally important role in the charac-terization of the Roemer rule. Imposing it indeed implies a considerable strengtheningof the rather weak idea of dominance compensation. This is summarized in the followinglemma:

Lemma 3. If a social ordering R defined on D satisfies UD, SI∗M and DC, then it alsosatisfies C.

Note that we described in the previous section how compensation C captures the mainintuition of the CO-approach. The basic incompatibility between the Roemer rule andthe Van de gaer rule can be summarized in the following lemma:

Lemma 4. Suppose m,n ≥ 2. A social ordering R defined on D cannot satisfy UD,SP, C and UN simultaneously.

Combining these various lemmas, we obtain the following message. If we want to keepUD, Itntf , SP, and DC, we will have to choose either SI

∗M , or SI

∗N . Choosing SI

∗M implies

choosing C and choosing SI∗N implies choosing UN, and even UC. The choice betweenboth impartiality axioms can therefore be seen as the choice between compensatingoutcomes via C, or compensating opportunity sets via UC, where the value of theopportunity sets is measured as the sum of the outcomes achieved by individuals withthe same type. This dichotomy is also the driving force behind the example given insection 2 (figure 4).

6 Extensions

The multiplicative extensions In the previous section, we have shown that boththe Roemer rule and the Van de gaer rule satisfy the invariance axiom Itntf . It is notobvious that this is an attractive idea. First, it implies an absolute approach to theevaluation of inequality of opportunity. Such an absolute approach is not the mostpopular in the literature on inequality measurement. Second, it implies a completeagnosticism about the ethical evaluation of differences between the outcomes related todifferent effort levels.The most obvious alternative to Itntf is to follow the dominant relative approach toinequality measurement: within a column we look at relative, rather than absolute

13

differences in outcomes. This means that we only allow for ratios of outcomes betweentypes with the same responsibility factors to be relevant. Consider Π the domain ofdiagonal matrices β ∈ Rn×n with strictly positive diagonal entries βii > 0. We thenpropose:

• Irnrf (invariance axiom)For all u, v ∈ D and for all β ∈ Π; uRv if and only if (uβ)R (vβ) .

Similar to Itntf , this invariance axiom Irnrf implies separability between effort levels SEN .Furthermore, Irnrf preserves the relative inequality between outcomes for a given effortlevel. Since this holds for all effort levels, it can be maintained that this does not changethe inequality of opportunity sets either, if one takes a relative perspective on inequality.With the necessary change in domain, we get:

Proposition 2. For each row in the table, there is only one social ordering R whichsatisfies the axioms indicated by •; it is the rule in the first column. In addition, +(resp. − ) means that the rule satisfies (violates) the corresponding axiom:

PD Irnrf SP DC SEM SI∗M SIM SI∗N SINproduct-Roemer • • • • • • + − •product-Van de gaer • • • • + − • • +

The flexible extensions We can go further and drop the assumption of non-comparability between columns. Again, this is easily done by exploiting the analogy withthe classical invariance requirements in the social choice literature. Define 1nm ∈ Rm×n,a matrix consisting of ones:

• Itftf (invariance axiom).For all u, v ∈ D and for all a ∈ R; uRv if and only if (u+ a1nm)R (v + a1nm) .

• Irfrf (invariance axiom).For all u, v ∈ D and for all b ∈ R++; uRv if and only if (bu)R (bv) .

Itftf (resp. Irfrf) does not imply separability between effort levels; separability axioms will

be imposed directly. Furthermore, Itftf (resp. Irfrf) only compares situations that keep the

absolute (resp. relative) differences between outcomes at all effort levels constant. Theseweaker axioms allow us to express a wide spectrum of opinions about inequalities thatare due to type and effort. The following proposition characterizes the Kolm-Atkinson-Sen (KAS) and Kolm-Pollak (KP) specification for the Roemer rule and the Van degaer rule:

Proposition 3. For each row in the table, there is only one family of social orderings Rwhich satisfies the axioms indicated by •; it is the family in the first column. In addition,+ (resp. − ) means that each rule in the family satisfies (violates) the correspondingaxiom:

14

PD Irfrf SP CON SE∗M SEM SE∗N SEN SI∗M SIM SI∗N SINKAS-R • • • • • • + • + •KAS-VDG • • • • • + • • • +

UD ItftfKP-R • • • • • • + • + •KP-VDG • • • • • + • • • +

R stands for Roemer and VDG for Van de gaer.

It remains true that the main differences between the (broad) Roemer rule and Vande gaer rule (and therefore also between the CO and the CS approaches to equality ofopportunity) are revealed by the impartiality axioms, but they are here reinforced byspecific separability axioms.We can conclude that the invariance axiom Itntf is crucial in explaining the sum-assumptionwhich is imposed both by Roemer and Van de gaer. Alternatives to these rules are read-ily available for those who do not like the (strong) implications of Itntf : absolute inequalitymeasurement within a column, i.e. for a given effort level, and complete neglect of dif-ferences in outcome levels between columns, i.e. for different effort levels. Of course,none of these alternatives counters the criticism that with one type the laissez-faireoutcomes should be respected. However, if one accepts the broader interpretation ofresponsibility-sensitivity in which the social evaluation also depends on the form of theopportunity sets, the increased flexibility gained by relaxing Itntf seems a definite im-provement. Note that by imposing even weaker invariance requirements than Itftf andIrfrf, many additional functional forms become possible.

7 Conclusion

Although the Roemer rule and the Van de gaer rule are formally similar, their basicintuition is very different. This difference goes deeper than a mere difference in normal-ization. The former exploits the idea of compensating for different outcomes, if thesedifferences follow from differences in non-responsibility or compensation factors. Thelatter focuses on the evaluation of opportunity sets and wants to make their “value”as equal as possible. As we have shown, both approaches are incompatible in general.While they lead to similar policy prescriptions in many cases (and to identical policyprescriptions if there is one type whose opportunity set is dominated by all the othertypes), this should not detract us from the differences in ethical inspiration. It would beinteresting to further explore these differences in concrete applications.14 More empiricalwork could be very fruitful in this regard.Both rules satisfy an invariance axiom which we have labeled Itntf . It is reflected in theiradditive specification. The axiom Itntf imposes that the social evaluation does not changeif the same constant is added to the outcomes of all types at a given effort level. Thisessentially implies an absolute approach to inequality measurement. Moreover, these

14Van de gaer et al. (2001) explore the justification of the rules in the context of the measurement ofintergenerational mobility. Schokkaert et al. (2002) calculate the optimal linear income tax using theserules (among others).

15

constants may differ at different effort levels, implying that differences between theoutcomes related to different effort levels do not matter. All these assumptions aredebatable. However, we have shown that introducing other invariance axioms leads torules of the Roemer rule and Van de gaer-family with a more flexible specification.The importance of this increased flexibility should not be underestimated. As has beennoted by Fleurbaey (1998), neither the Roemer rule nor the Van de gaer rule satisfythe traditional notion of equality of opportunity implying that the laissez-faire outcomeshould be respected in cases where there is only one type. This is one specific notion ofresponsibility-sensitive egalitarianism, however. It can also be argued that responsibilitymatters, i.e. that effort should be rewarded, while at the same time putting forwarddefinite ideas about how it should be rewarded and, more specifically, without acceptingthat the laissez-faire rewards to effort are necessarily ideal from an ethical point of view.This requires that the form of opportunity sets is evaluated, even if there is only onetype. The additive (or utilitarian) specification underlying the original Roemer rule andthe Van de gaer rule may not be the most attractive candidate for such an evaluationexercise. The more flexible specifications introducing a parameter of inequality aversioncan then be seen as a first step into the direction of a broader notion of responsibility-sensitive egalitarianism.

16

Appendix

Proof of lemmas

Proof of lemma 1 If a social ordering R defined on D satisfies UD, Itntf and SI∗N ,

then it also satisfies UN. Consider a matrix u ∈ D and two couples ij, ik in MN .Construct a matrix v by permuting elements ij and ik in matrix u. Using SI∗N wehave uIv. Add an arbitrary amount a ∈ R to the outcomes in the j-th column of bothmatrices u and v to get u and v ; using Itntf , we have u Iv . Permute the elements ijand ik in matrix u to get u and by SI∗N and transitivity of R, we have u Iv . Subtractthe same amount a from the elements in the j-th column of both matrices u and vto get u and v = v; from Itntf , reflexivity and transitivity of R, we have u Iv. By

construction, the matrix u is everywhere equal to v, except for outcomes u ji = vji −a

and u ki = vki + a. Hence, we may remove outcome mass within a row of a matrix v,

without changing social welfare, more precisely:

• Transfer principle.For all v ∈ D, for all ij, ik ∈ MN and for all a ∈ R; vIv , with v ji = vji + a,v ki = vki − a and v hg = vhg for all gh ∈MN\ {ij, ik} .

Repeatedly using this “transfer principle”, UN must hold, as required.

Proof of lemma 2 If a social ordering R defined on D satisfies UD, Itntf , SI∗N and

DC, then it also satisfies UC. From lemma 1, we know that UD, Itntf and SI∗N implies UN.

We prove that adding DC also implies UC. Suppose the antecedents of UC are true fortwo matrices u, v ∈ D and two types i, k in M , i.e. j∈N v

jk > j∈N u

jk ≥ j∈N u

ji >

j∈N vji , while u

hg = vhg for all gh /∈ G = iN ∪ kN . Construct two matrices u and v as

follows:

(i) v 1k = j∈N vjk (ii) u 1k = j∈N u

jk (iii) u 1i = j∈N u

ji (iv) v 1i = j∈N v

ji

(v) u hg = v hg = 0,∀gh = i1, k1 & gh ∈ G (vi) u hg = uhg & vhg = vhg ,∀gh /∈ G.

From UN, we have uIu and v Iv. But by construction, DC applies and we have u Rv .From transitivity also uRv holds.

Proof of lemma 3 If a social ordering R defined on D satisfies UD, SI∗M and DC,then it also satisfies C. Suppose the antecedents of C are true for u, v ∈ D and for ij, kjin MN , i.e. vjk > ujk ≥ uji > vji , while u

hg = vhg for all other gh = ij, kj. Construct

two matrices u and v by reranking all columns different from j, either increasinglyor decreasingly depending on whether i < k or i > k. From SI∗M , we have uIu andvIv . By construction, we can apply DC to get u Rv and thus also uRv must hold, asrequired.

Proof of lemma 4 Suppose m,n ≥ 2. A social ordering R defined on D cannotsatisfy UD, SP, C and UN simultaneously. First, consider the following four matrices

u =3 83 6

, v =2 84 6

, u =2 104 5

, v =3 93 6

17

Using C, we have uRv and vRu . Using UN, we have u Iv . Everything together, weobtain by transitivity that uRv must hold, which contradicts SP. We can always embedthese four matrices in larger ones belonging to Rm×n, which completes the proof.15

Proof of proposition 1

A. Independence of the axioms We first show that, for both rules, dropping theaxioms one by one results in at least one new rule, which satisfies all, but the droppedaxiom. We also show that, restricting the domain, might result in new rules.

A1. Roemer rule

• restrict to domain D (of column-ordered matrices):16 the Van de gaer rule satisfiesall axioms (as SI∗M becomes empty)

• drop Itntf : consider the ”extensive” leximin rule, defined as the standard leximinrule RL, applied to vectors

u11, u21, . . . , u

n1 ;u

12, . . . , u

n2 ; . . . ;u

1m, . . . , u

nm ∈ Rmn.

• drop SP: consider the original “maxmin”-version of the Roemer rule.• drop DC: consider the “extensive” utilitarian rule RU , defined as the standardutilitarian rule, applied to vectors

u11, u21, . . . , u

n1 ;u

12, . . . , u

n2 ; . . . ;u

1m, . . . , u

nm ∈ Rmn.

• drop SEM : consider the following rerank operation:

1. first, change u into u, by reordering the outcomes in each column in anincreasing way, and

2. second, change u into u, by reordering the columns such that (i) the firstrow is ordered increasingly (from left to right) and (ii) if ex-aequo’s arepresent, use the outcomes in the second (or further) rows to decide about theranking.17

Now, apply the standard leximin rule RL in a serial way to the columns, startingwith the first column, or for all u, v ∈ D, we have

uRv ⇔ ujRLvj , ∀j ∈ N , or,∃k ∈ N s.t. ukPLvk and (if k > 1) ujILvj ,∀j = 1, . . . , k − 1.

15This result also holds (i) on the positive domain, and/or (ii) without completeness of the socialrelation.16Notice that the axioms have to be adapted slightly, to ensure that transformations of utility vectors

are within the new domain.17For example,

u =

0 3 27 6 32 0 0

→ u =

0 0 02 3 27 6 3

→ 0 0 02 2 37 3 6

→ u =

0 0 02 2 33 7 6

.

18

• SI∗M : consider the Van de gaer rule• SIN : adapt the Roemer rule using (strictly positive) weights when summing (themins).

A2. Van de gaer rule

• restrict to domain D (of column-ordered matrices):18 the Roemer rule satisfies allaxioms (as SI∗N becomes empty here)

• drop Itntf : consider the ”extensive” leximin rule, defined in A1.• drop SP: consider a “maxmin”-version of the Van de gaer rule.• drop DC: consider the “extensive” utilitarian rule, defined in A1.• drop SI∗N : consider the Roemer rule.• drop SIM : adapt the Van de gaer rule using weights (∈ N0) when leximinning (thesums).19

B. Characterization of the Roemer rule It is easy to verify existence: the Roe-mer rule satisfies all axioms UD, Itntf , SP, DC, SEM , SI

∗M and SIN . To prove uniqueness,

we need two simple lemmas, which deal with matrices u belonging to

D (r) = u ∈ D and, if r > 1, u1 = . . . = ur−1 = (0, . . . , 0) ∈ Rn holds .

We do not prove both lemmas, as they directly follow from the axioms (sometimes usedrepeatedly), indicated at the beginning of each procedure step.

Lemma P1 (r). Consider the procedure P1 (r), with r ∈M , which transforms matricesa, b ∈ D (r) into g, h ∈ D (r):

• (SIN) rerank the columns in a, b such that the r-th row is ordered increasingly;call c, d the resulting matrices.

• (SEM) if r > 1, define a vector δ = min c1r, d1r , . . . ,min (cnr , d

nr ) ∈ Rn and add

it to the rows 1, . . . , r − 1 of matrices c, d; call e, f the resulting matrices.

• (Itntf ) subtract the same δ = min c1r, d1r , . . . ,min (cnr , d

nr ) ∈ Rn from all rows of

matrices e, f ; call g, h the resulting matrices.

If a social orderingR satisfies UD, SIN , SEM , and Itntf , then gRh⇒ aRb and gPh⇒ aPb.

Lemma P2 (r). Consider the procedure P2 (r), with r ∈M , which transforms matricesa, c ∈ D (r) into b, c ∈ D (r):

18Notice that the axioms have to be adapted slightly, to ensure that transformations of utility vectorsare within the new domain.19As it is rather unusual, weighting the leximin rule with positive integers has to be understood as

applying the leximin rule to numbers, which are replicated by its weight. For example comparing (1, 2, 3)with (2, 1, 3) and using (1, 2, 1) as weighting vector, amounts to comparing (1, 2, 2, 3) and (2, 1, 1, 3) viathe leximin rule.

19

• define ε = 1 + max 0, j∈N bjr+1 − j∈N a

jr+1 ; thus ε > 0.

• (DC) whenever ajr > 0 for some j ∈ N , perform a series of inequitable transfers:reduce ajr to zero and add ε to a

jr+1, . . . , a

jm; call b the resulting matrix and notice

that b ∈ D (r + 1).

If a social ordering R satisfies UD, and DC, then bRc⇒ aRc and bPc⇒ aPc.

Proof. We are now ready to prove uniqueness. Let R be the Roemer social orderingand let R be an ordering satisfying all axioms. We have to prove R = R. Therefore, itsuffices to prove for all u, v ∈ D, that (i) uRv ⇒ uRv, and, (ii) uPv ⇒ uPv. Considertwo matrices u, v ∈ D for which uRv holds (the case uPv is analogous, by replacing“R” by “P” in the sequel). We have to show that uRv holds. First, rerank the columnelements in u, v in an increasing way, such that the resulting matrices a, b ∈ D (1). UsingSI∗M , we have to prove aRb.Step 1 (if m > 1): In case j∈N a

j1 = j∈N b

j1, use procedure P1 (1) a finite number of

times until the resulting matrices d, e both belong to D (2). In case j∈N aj1 > j∈N b

j1,

use procedure P1 (1) a finite number of times until the second (called e) of the resultingmatrices c, e belongs to D (2), followed by procedure P2 (1) applied to c, e to obtainmatrices d, e which both belong to D (2).20 Using lemmas P1 (1) and P2 (1), it sufficesto prove dRe.Step 2 (if m > 2): We apply the sequence of procedures described in step 1, but now tothe second row.Step m: After m − 1 steps, we have to matrices w, x ∈ D (m). Applying procedureP1 (m) a finite number of times leads to matrices y, z ∈ D (m) with ym ≥ zm. Usinglemma P1 (m) we have to prove yRz, which follows from SP.

C. Characterization of the Van de gaer rule It is easy to verify existence: theVan de gaer rule satisfies all axioms UD, Itntf , SP, DC, SI

∗N and SIM . We prove uniqueness.

Let R be the Van de gaer social ordering and let R be an ordering satisfying all axioms.We have to prove R = R.From lemma 1, R has to satisfy UN. Using UN and transitivity, uRv holds if and onlyif uRv, with u (and v) defined as

u =

u11 = j∈N uj1 0 . . . 0

. . . . . . . . .

u1m = j∈N ujm 0 . . . 0

.Consider two matrices u, v ∈ D and define an orderingR◦, defined onRm, as uRv (⇔ uRv)⇔u1R◦v1. The induced ordering R◦ is well-defined, and inherits (from the properties SP,SIM and DC for R) the strong Pareto principle, Suppes’ indifference, and a slightlystronger version of Hammond Equity, suitably redefined on the new domain Rm. Butthen, R◦ has to be the leximin rule (see, for instance, Bossert and Weymark, 1996,theorem 15), and, by definition of R◦, we have R = R.

20Notice that the case j∈N aj1 < j∈N b

j1 is not possible as it would contradict uRv.

20

Proof of proposition 2

First, consider a social ordering R satisfying PD and Irnrf . Define a social ordering R◦

such that for all u, v ∈ Rm×n++ : lnuR◦ ln v ⇔ uRv, with lnx equal to the matrix lnxjifor x = u, v. It is easy to see that R◦ has to satisfy UD and Itntf . Second, if we imposeone of the axioms SP, DC, SEM , SI∗M , SIM , SI

∗N or SIN on R, then the same axiom

must hold for R◦. Finally, we can use proposition 1 to characterize R◦: depending onwhether we additionally impose the set SP, DC, SEM , SI∗M , SIN or the set SP, DC,SIM , SI∗N , we obtain that R

◦ must be the Roemer rule (resp. the Van de gaer rule).By definition of both rules and by definition of R◦, we get that R must be the Roemerrule (resp. the Van de gaer rule), but applied to logarithmically transformed matrices;this is equal to the product-Roemer rule (resp. product-Van de gaer rule) applied to theoriginal matrices u, v ∈ Rm×n++ .

Proof of proposition 3

We prove the characterization for the KAS-Roemer-family; the other cases are analo-gous. All orderings in the family satisfy the axioms PD, Irfrf, SP, CON, SEM , SE

∗N , SI

∗M ,

and SIN . The other way around, consider a social ordering R satisfying these axioms.Using CON and SE∗N , this social ordering can be represented by continuous functionsf and g, such that for all u, v ∈ D:

uRv ⇔ f g u1 , . . . , g (un) ≥ f g v1 , . . . , g (vn) .

Via SP, SIN and SE∗N for the function f and via SP, SEM and SI∗M for the function g, wecan represent both f and g by continuous and strictly monotone functions, respectivelyf1, f2 and g1, g2:

f (a1, . . . , an) = (f1)−1j∈N

f2 (aj)

and g (b1, . . . , bm) = (g1)−1

i∈Mg2 (bi) .

Finally, Irfrf imposes homotheticity, which results in the KAS-class of functions, as re-quired.

21

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